The slowing down of energetic ions in materials is determined by the nuclear and electronic stopping powers. Both of these have been studied extensively for ordinary-matter ions. For antiprotons, however, there are numerous studies of the electronic stopping power, but none of the nuclear stopping power. Here, we use quantum-chemical methods to calculate interparticle potentials between antiprotons and different atoms, and derive from these the nuclear stopping power of antiprotons in solids. The results show that the antiproton nuclear stopping powers are much stronger than those of protons, and can also be stronger than the electronic stopping power at the lowest energies. The interparticle potentials are also implemented in a molecular dynamics ion range calculation code, which allows us to simulate antiproton transmission through degrader foil materials. Foil transmission simulations carried out at experimentally relevant conditions show that the choice of antiproton-atom interaction model has a large effect on the predicted yield of antiprotons slowed down to low (a few keV) energies.